* Its most important design principle is to completely hide the inner
* working of that other package from the user of GiNaC. It must either
* provide implementation of arithmetic operators and numerical evaluation
- * of special functions or implement the interface to the bignum package.
- *
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ * of special functions or implement the interface to the bignum package. */
+
+/*
+ * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
+#include "config.h"
+
#include <vector>
#include <stdexcept>
+#include <string>
+
+#if defined(HAVE_SSTREAM)
+#include <sstream>
+#elif defined(HAVE_STRSTREAM)
+#include <strstream>
+#else
+#error Need either sstream or strstream
+#endif
#include "numeric.h"
#include "ex.h"
-#include "config.h"
+#include "archive.h"
+#include "debugmsg.h"
+#include "utils.h"
// CLN should not pollute the global namespace, hence we include it here
-// instead of in some header file where it would propagate to other parts:
+// instead of in some header file where it would propagate to other parts.
+// Also, we only need a subset of CLN, so we don't include the complete cln.h:
#ifdef HAVE_CLN_CLN_H
-#include <CLN/cln.h>
-#else
-#include <cln.h>
-#endif
+#include <cln/cl_output.h>
+#include <cln/cl_integer_io.h>
+#include <cln/cl_integer_ring.h>
+#include <cln/cl_rational_io.h>
+#include <cln/cl_rational_ring.h>
+#include <cln/cl_lfloat_class.h>
+#include <cln/cl_lfloat_io.h>
+#include <cln/cl_real_io.h>
+#include <cln/cl_real_ring.h>
+#include <cln/cl_complex_io.h>
+#include <cln/cl_complex_ring.h>
+#include <cln/cl_numtheory.h>
+#else // def HAVE_CLN_CLN_H
+#include <cl_output.h>
+#include <cl_integer_io.h>
+#include <cl_integer_ring.h>
+#include <cl_rational_io.h>
+#include <cl_rational_ring.h>
+#include <cl_lfloat_class.h>
+#include <cl_lfloat_io.h>
+#include <cl_real_io.h>
+#include <cl_real_ring.h>
+#include <cl_complex_io.h>
+#include <cl_complex_ring.h>
+#include <cl_numtheory.h>
+#endif // def HAVE_CLN_CLN_H
+
+#ifndef NO_NAMESPACE_GINAC
+namespace GiNaC {
+#endif // ndef NO_NAMESPACE_GINAC
// linker has no problems finding text symbols for numerator or denominator
//#define SANE_LINKER
+GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
+
//////////
// default constructor, destructor, copy constructor assignment
// operator and helpers
{
debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
value = new cl_N;
- *value=cl_I(0);
+ *value = cl_I(0);
calchash();
- setflag(status_flags::evaluated|
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
status_flags::hash_calculated);
}
destroy(0);
}
-numeric::numeric(numeric const & other)
+numeric::numeric(const numeric & other)
{
debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
copy(other);
}
-numeric const & numeric::operator=(numeric const & other)
+const numeric & numeric::operator=(const numeric & other)
{
debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
if (this != &other) {
// protected
-void numeric::copy(numeric const & other)
+void numeric::copy(const numeric & other)
{
basic::copy(other);
value = new cl_N(*other.value);
{
debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
// Not the whole int-range is available if we don't cast to long
- // first. This is due to the behaviour of the cl_I-ctor, which
+ // first. This is due to the behaviour of the cl_I-ctor, which
// emphasizes efficiency:
value = new cl_I((long) i);
calchash();
status_flags::hash_calculated);
}
+
numeric::numeric(unsigned int i) : basic(TINFO_numeric)
{
debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
// Not the whole uint-range is available if we don't cast to ulong
- // first. This is due to the behaviour of the cl_I-ctor, which
+ // first. This is due to the behaviour of the cl_I-ctor, which
// emphasizes efficiency:
value = new cl_I((unsigned long)i);
calchash();
status_flags::hash_calculated);
}
+
numeric::numeric(long i) : basic(TINFO_numeric)
{
debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
status_flags::hash_calculated);
}
+
numeric::numeric(unsigned long i) : basic(TINFO_numeric)
{
debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
status_flags::hash_calculated);
}
+
numeric::numeric(double d) : basic(TINFO_numeric)
{
debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
status_flags::hash_calculated);
}
-numeric::numeric(char const *s) : basic(TINFO_numeric)
-{ // MISSING: treatment of complex and ints and rationals.
+
+numeric::numeric(const char *s) : basic(TINFO_numeric)
+{ // MISSING: treatment of complex numbers
debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
if (strchr(s, '.'))
value = new cl_LF(s);
else
- value = new cl_I(s);
+ value = new cl_R(s);
calchash();
setflag(status_flags::evaluated|
status_flags::hash_calculated);
/** Ctor from CLN types. This is for the initiated user or internal use
* only. */
-numeric::numeric(cl_N const & z) : basic(TINFO_numeric)
+numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
{
debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
value = new cl_N(z);
status_flags::hash_calculated);
}
+//////////
+// archiving
+//////////
+
+/** Construct object from archive_node. */
+numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+{
+ debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
+ value = new cl_N;
+
+ // Read number as string
+ string str;
+ if (n.find_string("number", str)) {
+#ifdef HAVE_SSTREAM
+ istringstream s(str);
+#else
+ istrstream s(str.c_str(), str.size() + 1);
+#endif
+ cl_idecoded_float re, im;
+ char c;
+ s.get(c);
+ switch (c) {
+ case 'R': // Integer-decoded real number
+ s >> re.sign >> re.mantissa >> re.exponent;
+ *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
+ break;
+ case 'C': // Integer-decoded complex number
+ s >> re.sign >> re.mantissa >> re.exponent;
+ s >> im.sign >> im.mantissa >> im.exponent;
+ *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
+ im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
+ break;
+ default: // Ordinary number
+ s.putback(c);
+ s >> *value;
+ break;
+ }
+ }
+ calchash();
+ setflag(status_flags::evaluated|
+ status_flags::hash_calculated);
+}
+
+/** Unarchive the object. */
+ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
+{
+ return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
+}
+
+/** Archive the object. */
+void numeric::archive(archive_node &n) const
+{
+ inherited::archive(n);
+
+ // Write number as string
+#ifdef HAVE_SSTREAM
+ ostringstream s;
+#else
+ char buf[1024];
+ ostrstream s(buf, 1024);
+#endif
+ if (this->is_crational())
+ s << *value;
+ else {
+ // Non-rational numbers are written in an integer-decoded format
+ // to preserve the precision
+ if (this->is_real()) {
+ cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
+ s << "R";
+ s << re.sign << " " << re.mantissa << " " << re.exponent;
+ } else {
+ cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
+ cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
+ s << "C";
+ s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
+ s << im.sign << " " << im.mantissa << " " << im.exponent;
+ }
+ }
+#ifdef HAVE_SSTREAM
+ n.add_string("number", s.str());
+#else
+ s << ends;
+ string str(buf);
+ n.add_string("number", str);
+#endif
+}
+
//////////
// functions overriding virtual functions from bases classes
//////////
return new numeric(*this);
}
-// The method printraw doesn't do much, it simply uses CLN's operator<<() for
-// output, which is ugly but reliable. Examples:
-// 2+2i
-void numeric::printraw(ostream & os) const
+
+/** Helper function to print a real number in a nicer way than is CLN's
+ * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
+ * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
+ * long as it only uses cl_LF and no other floating point types.
+ *
+ * @see numeric::print() */
+void print_real_number(ostream & os, const cl_R & num)
{
- debugmsg("numeric printraw", LOGLEVEL_PRINT);
- os << "numeric(" << *value << ")";
+ cl_print_flags ourflags;
+ if (::instanceof(num, ::cl_RA_ring)) {
+ // case 1: integer or rational, nothing special to do:
+ ::print_real(os, ourflags, num);
+ } else {
+ // case 2: float
+ // make CLN believe this number has default_float_format, so it prints
+ // 'E' as exponent marker instead of 'L':
+ ourflags.default_float_format = ::cl_float_format(The(cl_F)(num));
+ ::print_real(os, ourflags, num);
+ }
+ return;
}
-// The method print adds to the output so it blends more consistently together
-// with the other routines and produces something compatible to Maple input.
+/** This method adds to the output so it blends more consistently together
+ * with the other routines and produces something compatible to ginsh input.
+ *
+ * @see print_real_number() */
void numeric::print(ostream & os, unsigned upper_precedence) const
{
debugmsg("numeric print", LOGLEVEL_PRINT);
- if (is_real()) {
+ if (this->is_real()) {
// case 1, real: x or -x
- if ((precedence<=upper_precedence) && (!is_pos_integer())) {
- os << "(" << *value << ")";
+ if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
+ os << "(";
+ print_real_number(os, The(cl_R)(*value));
+ os << ")";
} else {
- os << *value;
+ print_real_number(os, The(cl_R)(*value));
}
} else {
// case 2, imaginary: y*I or -y*I
- if (realpart(*value) == 0) {
- if ((precedence<=upper_precedence) && (imagpart(*value) < 0)) {
- if (imagpart(*value) == -1) {
+ if (::realpart(*value) == 0) {
+ if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
+ if (::imagpart(*value) == -1) {
os << "(-I)";
} else {
- os << "(" << imagpart(*value) << "*I)";
+ os << "(";
+ print_real_number(os, The(cl_R)(::imagpart(*value)));
+ os << "*I)";
}
} else {
- if (imagpart(*value) == 1) {
+ if (::imagpart(*value) == 1) {
os << "I";
} else {
- if (imagpart (*value) == -1) {
+ if (::imagpart (*value) == -1) {
os << "-I";
} else {
- os << imagpart(*value) << "*I";
+ print_real_number(os, The(cl_R)(::imagpart(*value)));
+ os << "*I";
}
}
}
} else {
// case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
- if (precedence <= upper_precedence) os << "(";
- os << realpart(*value);
- if (imagpart(*value) < 0) {
- if (imagpart(*value) == -1) {
+ if (precedence <= upper_precedence)
+ os << "(";
+ print_real_number(os, The(cl_R)(::realpart(*value)));
+ if (::imagpart(*value) < 0) {
+ if (::imagpart(*value) == -1) {
os << "-I";
} else {
- os << imagpart(*value) << "*I";
+ print_real_number(os, The(cl_R)(::imagpart(*value)));
+ os << "*I";
}
} else {
- if (imagpart(*value) == 1) {
+ if (::imagpart(*value) == 1) {
os << "+I";
} else {
- os << "+" << imagpart(*value) << "*I";
+ os << "+";
+ print_real_number(os, The(cl_R)(::imagpart(*value)));
+ os << "*I";
}
}
- if (precedence <= upper_precedence) os << ")";
+ if (precedence <= upper_precedence)
+ os << ")";
+ }
+ }
+}
+
+
+void numeric::printraw(ostream & os) const
+{
+ // The method printraw doesn't do much, it simply uses CLN's operator<<()
+ // for output, which is ugly but reliable. e.g: 2+2i
+ debugmsg("numeric printraw", LOGLEVEL_PRINT);
+ os << "numeric(" << *value << ")";
+}
+
+
+void numeric::printtree(ostream & os, unsigned indent) const
+{
+ debugmsg("numeric printtree", LOGLEVEL_PRINT);
+ os << string(indent,' ') << *value
+ << " (numeric): "
+ << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
+ << ", flags=" << flags << endl;
+}
+
+
+void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
+{
+ debugmsg("numeric print csrc", LOGLEVEL_PRINT);
+ ios::fmtflags oldflags = os.flags();
+ os.setf(ios::scientific);
+ if (this->is_rational() && !this->is_integer()) {
+ if (compare(_num0()) > 0) {
+ os << "(";
+ if (type == csrc_types::ctype_cl_N)
+ os << "cl_F(\"" << numer().evalf() << "\")";
+ else
+ os << numer().to_double();
+ } else {
+ os << "-(";
+ if (type == csrc_types::ctype_cl_N)
+ os << "cl_F(\"" << -numer().evalf() << "\")";
+ else
+ os << -numer().to_double();
}
+ os << "/";
+ if (type == csrc_types::ctype_cl_N)
+ os << "cl_F(\"" << denom().evalf() << "\")";
+ else
+ os << denom().to_double();
+ os << ")";
+ } else {
+ if (type == csrc_types::ctype_cl_N)
+ os << "cl_F(\"" << evalf() << "\")";
+ else
+ os << to_double();
}
+ os.flags(oldflags);
}
+
bool numeric::info(unsigned inf) const
{
switch (inf) {
- case info_flags::numeric:
- case info_flags::polynomial:
- case info_flags::rational_function:
+ case info_flags::numeric:
+ case info_flags::polynomial:
+ case info_flags::rational_function:
+ return true;
+ case info_flags::real:
+ return is_real();
+ case info_flags::rational:
+ case info_flags::rational_polynomial:
+ return is_rational();
+ case info_flags::crational:
+ case info_flags::crational_polynomial:
+ return is_crational();
+ case info_flags::integer:
+ case info_flags::integer_polynomial:
+ return is_integer();
+ case info_flags::cinteger:
+ case info_flags::cinteger_polynomial:
+ return is_cinteger();
+ case info_flags::positive:
+ return is_positive();
+ case info_flags::negative:
+ return is_negative();
+ case info_flags::nonnegative:
+ return !is_negative();
+ case info_flags::posint:
+ return is_pos_integer();
+ case info_flags::negint:
+ return is_integer() && is_negative();
+ case info_flags::nonnegint:
+ return is_nonneg_integer();
+ case info_flags::even:
+ return is_even();
+ case info_flags::odd:
+ return is_odd();
+ case info_flags::prime:
+ return is_prime();
+ case info_flags::algebraic:
+ return !is_real();
+ }
+ return false;
+}
+
+/** Disassemble real part and imaginary part to scan for the occurrence of a
+ * single number. Also handles the imaginary unit. It ignores the sign on
+ * both this and the argument, which may lead to what might appear as funny
+ * results: (2+I).has(-2) -> true. But this is consistent, since we also
+ * would like to have (-2+I).has(2) -> true and we want to think about the
+ * sign as a multiplicative factor. */
+bool numeric::has(const ex & other) const
+{
+ if (!is_exactly_of_type(*other.bp, numeric))
+ return false;
+ const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
+ if (this->is_equal(o) || this->is_equal(-o))
return true;
- case info_flags::real:
- return is_real();
- case info_flags::rational:
- case info_flags::rational_polynomial:
- return is_rational();
- case info_flags::integer:
- case info_flags::integer_polynomial:
- return is_integer();
- case info_flags::positive:
- return is_positive();
- case info_flags::negative:
- return is_negative();
- case info_flags::nonnegative:
- return compare(numZERO())>=0;
- case info_flags::posint:
- return is_pos_integer();
- case info_flags::negint:
- return is_integer() && (compare(numZERO())<0);
- case info_flags::nonnegint:
- return is_nonneg_integer();
- case info_flags::even:
- return is_even();
- case info_flags::odd:
- return is_odd();
- case info_flags::prime:
- return is_prime();
+ if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
+ return (this->real().is_equal(o) || this->imag().is_equal(o) ||
+ this->real().is_equal(-o) || this->imag().is_equal(-o));
+ else {
+ if (o.is_equal(I)) // e.g scan for I in 42*I
+ return !this->is_real();
+ if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
+ return (this->real().has(o*I) || this->imag().has(o*I) ||
+ this->real().has(-o*I) || this->imag().has(-o*I));
}
return false;
}
+
+/** Evaluation of numbers doesn't do anything at all. */
+ex numeric::eval(int level) const
+{
+ // Warning: if this is ever gonna do something, the ex ctors from all kinds
+ // of numbers should be checking for status_flags::evaluated.
+ return this->hold();
+}
+
+
/** Cast numeric into a floating-point object. For example exact numeric(1) is
* returned as a 1.0000000000000000000000 and so on according to how Digits is
* currently set.
*
* @param level ignored, but needed for overriding basic::evalf.
- * @return an ex-handle to a numeric. */
+ * @return an ex-handle to a numeric. */
ex numeric::evalf(int level) const
{
// level can safely be discarded for numeric objects.
- return numeric(cl_float(1.0, cl_default_float_format) * (*value)); // -> CLN
+ return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
}
// protected
-int numeric::compare_same_type(basic const & other) const
+/** Implementation of ex::diff() for a numeric. It always returns 0.
+ *
+ * @see ex::diff */
+ex numeric::derivative(const symbol & s) const
+{
+ return _ex0();
+}
+
+
+int numeric::compare_same_type(const basic & other) const
{
- ASSERT(is_exactly_of_type(other, numeric));
- numeric const & o = static_cast<numeric &>(const_cast<basic &>(other));
+ GINAC_ASSERT(is_exactly_of_type(other, numeric));
+ const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
if (*value == *o.value) {
return 0;
return compare(o);
}
-bool numeric::is_equal_same_type(basic const & other) const
+
+bool numeric::is_equal_same_type(const basic & other) const
{
- ASSERT(is_exactly_of_type(other,numeric));
- numeric const *o = static_cast<numeric const *>(&other);
+ GINAC_ASSERT(is_exactly_of_type(other,numeric));
+ const numeric *o = static_cast<const numeric *>(&other);
- return is_equal(*o);
+ return this->is_equal(*o);
+}
+
+unsigned numeric::calchash(void) const
+{
+ return (hashvalue=cl_equal_hashcode(*value) | 0x80000000U);
+ /*
+ cout << *value << "->" << hashvalue << endl;
+ hashvalue=HASHVALUE_NUMERIC+1000U;
+ return HASHVALUE_NUMERIC+1000U;
+ */
}
/*
/** Numerical addition method. Adds argument to *this and returns result as
* a new numeric object. */
-numeric numeric::add(numeric const & other) const
+numeric numeric::add(const numeric & other) const
{
return numeric((*value)+(*other.value));
}
/** Numerical subtraction method. Subtracts argument from *this and returns
* result as a new numeric object. */
-numeric numeric::sub(numeric const & other) const
+numeric numeric::sub(const numeric & other) const
{
return numeric((*value)-(*other.value));
}
/** Numerical multiplication method. Multiplies *this and argument and returns
* result as a new numeric object. */
-numeric numeric::mul(numeric const & other) const
+numeric numeric::mul(const numeric & other) const
{
- static const numeric * numONEp=&numONE();
- if (this==numONEp) {
+ static const numeric * _num1p=&_num1();
+ if (this==_num1p) {
return other;
- } else if (&other==numONEp) {
+ } else if (&other==_num1p) {
return *this;
}
return numeric((*value)*(*other.value));
* a new numeric object.
*
* @exception overflow_error (division by zero) */
-numeric numeric::div(numeric const & other) const
+numeric numeric::div(const numeric & other) const
{
- if (zerop(*other.value))
+ if (::zerop(*other.value))
throw (std::overflow_error("division by zero"));
return numeric((*value)/(*other.value));
}
-numeric numeric::power(numeric const & other) const
+numeric numeric::power(const numeric & other) const
{
- static const numeric * numONEp=&numONE();
- if (&other==numONEp) {
+ static const numeric * _num1p = &_num1();
+ if (&other==_num1p)
return *this;
+ if (::zerop(*value)) {
+ if (::zerop(*other.value))
+ throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
+ else if (::zerop(::realpart(*other.value)))
+ throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
+ else if (::minusp(::realpart(*other.value)))
+ throw (std::overflow_error("numeric::eval(): division by zero"));
+ else
+ return _num0();
}
- if (zerop(*value) && other.is_real() && minusp(realpart(*other.value)))
- throw (std::overflow_error("division by zero"));
- return numeric(expt(*value,*other.value));
+ return numeric(::expt(*value,*other.value));
}
/** Inverse of a number. */
numeric numeric::inverse(void) const
{
- return numeric(recip(*value)); // -> CLN
+ return numeric(::recip(*value)); // -> CLN
}
-numeric const & numeric::add_dyn(numeric const & other) const
+const numeric & numeric::add_dyn(const numeric & other) const
{
- return static_cast<numeric const &>((new numeric((*value)+(*other.value)))->
+ return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
setflag(status_flags::dynallocated));
}
-numeric const & numeric::sub_dyn(numeric const & other) const
+const numeric & numeric::sub_dyn(const numeric & other) const
{
- return static_cast<numeric const &>((new numeric((*value)-(*other.value)))->
+ return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
setflag(status_flags::dynallocated));
}
-numeric const & numeric::mul_dyn(numeric const & other) const
+const numeric & numeric::mul_dyn(const numeric & other) const
{
- static const numeric * numONEp=&numONE();
- if (this==numONEp) {
+ static const numeric * _num1p=&_num1();
+ if (this==_num1p) {
return other;
- } else if (&other==numONEp) {
+ } else if (&other==_num1p) {
return *this;
}
- return static_cast<numeric const &>((new numeric((*value)*(*other.value)))->
+ return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
setflag(status_flags::dynallocated));
}
-numeric const & numeric::div_dyn(numeric const & other) const
+const numeric & numeric::div_dyn(const numeric & other) const
{
- if (zerop(*other.value))
+ if (::zerop(*other.value))
throw (std::overflow_error("division by zero"));
- return static_cast<numeric const &>((new numeric((*value)/(*other.value)))->
+ return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
setflag(status_flags::dynallocated));
}
-numeric const & numeric::power_dyn(numeric const & other) const
+const numeric & numeric::power_dyn(const numeric & other) const
{
- static const numeric * numONEp=&numONE();
- if (&other==numONEp) {
+ static const numeric * _num1p=&_num1();
+ if (&other==_num1p)
return *this;
+ if (::zerop(*value)) {
+ if (::zerop(*other.value))
+ throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
+ else if (::zerop(::realpart(*other.value)))
+ throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
+ else if (::minusp(::realpart(*other.value)))
+ throw (std::overflow_error("numeric::eval(): division by zero"));
+ else
+ return _num0();
}
- // The ifs are only a workaround for a bug in CLN. It gets stuck otherwise:
- if ( !other.is_integer() &&
- other.is_rational() &&
- (*this).is_nonneg_integer() ) {
- if ( !zerop(*value) ) {
- return static_cast<numeric const &>((new numeric(exp(*other.value * log(*value))))->
- setflag(status_flags::dynallocated));
- } else {
- if ( !zerop(*other.value) ) { // 0^(n/m)
- return static_cast<numeric const &>((new numeric(0))->
- setflag(status_flags::dynallocated));
- } else { // raise FPE (0^0 requested)
- return static_cast<numeric const &>((new numeric(1/(*other.value)))->
- setflag(status_flags::dynallocated));
- }
- }
- } else { // default -> CLN
- return static_cast<numeric const &>((new numeric(expt(*value,*other.value)))->
- setflag(status_flags::dynallocated));
- }
+ return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
+ setflag(status_flags::dynallocated));
}
-numeric const & numeric::operator=(int i)
+const numeric & numeric::operator=(int i)
{
return operator=(numeric(i));
}
-numeric const & numeric::operator=(unsigned int i)
+const numeric & numeric::operator=(unsigned int i)
{
return operator=(numeric(i));
}
-numeric const & numeric::operator=(long i)
+const numeric & numeric::operator=(long i)
{
return operator=(numeric(i));
}
-numeric const & numeric::operator=(unsigned long i)
+const numeric & numeric::operator=(unsigned long i)
{
return operator=(numeric(i));
}
-numeric const & numeric::operator=(double d)
+const numeric & numeric::operator=(double d)
{
return operator=(numeric(d));
}
-numeric const & numeric::operator=(char const * s)
+const numeric & numeric::operator=(const char * s)
{
return operator=(numeric(s));
}
+/** Return the complex half-plane (left or right) in which the number lies.
+ * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
+ * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
+ *
+ * @see numeric::compare(const numeric & other) */
+int numeric::csgn(void) const
+{
+ if (this->is_zero())
+ return 0;
+ if (!::zerop(::realpart(*value))) {
+ if (::plusp(::realpart(*value)))
+ return 1;
+ else
+ return -1;
+ } else {
+ if (::plusp(::imagpart(*value)))
+ return 1;
+ else
+ return -1;
+ }
+}
+
/** This method establishes a canonical order on all numbers. For complex
* numbers this is not possible in a mathematically consistent way but we need
* to establish some order and it ought to be fast. So we simply define it
- * similar to Maple's csgn. */
-int numeric::compare(numeric const & other) const
+ * to be compatible with our method csgn.
+ *
+ * @return csgn(*this-other)
+ * @see numeric::csgn(void) */
+int numeric::compare(const numeric & other) const
{
// Comparing two real numbers?
- if (is_real() && other.is_real())
+ if (this->is_real() && other.is_real())
// Yes, just compare them
- return cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
+ return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
else {
// No, first compare real parts
- cl_signean real_cmp = cl_compare(realpart(*value), realpart(*other.value));
+ cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
if (real_cmp)
return real_cmp;
- return cl_compare(imagpart(*value), imagpart(*other.value));
+ return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
}
}
-bool numeric::is_equal(numeric const & other) const
+bool numeric::is_equal(const numeric & other) const
{
return (*value == *other.value);
}
/** True if object is zero. */
bool numeric::is_zero(void) const
{
- return zerop(*value); // -> CLN
+ return ::zerop(*value); // -> CLN
}
/** True if object is not complex and greater than zero. */
bool numeric::is_positive(void) const
{
- if (is_real()) {
- return plusp(The(cl_R)(*value)); // -> CLN
- }
+ if (this->is_real())
+ return ::plusp(The(cl_R)(*value)); // -> CLN
return false;
}
/** True if object is not complex and less than zero. */
bool numeric::is_negative(void) const
{
- if (is_real()) {
- return minusp(The(cl_R)(*value)); // -> CLN
- }
+ if (this->is_real())
+ return ::minusp(The(cl_R)(*value)); // -> CLN
return false;
}
/** True if object is a non-complex integer. */
bool numeric::is_integer(void) const
{
- return (bool)instanceof(*value, cl_I_ring); // -> CLN
+ return ::instanceof(*value, ::cl_I_ring); // -> CLN
}
/** True if object is an exact integer greater than zero. */
bool numeric::is_pos_integer(void) const
{
- return (is_integer() &&
- plusp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN
}
/** True if object is an exact integer greater or equal zero. */
bool numeric::is_nonneg_integer(void) const
{
- return (is_integer() &&
- !minusp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN
}
/** True if object is an exact even integer. */
bool numeric::is_even(void) const
{
- return (is_integer() &&
- evenp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN
}
/** True if object is an exact odd integer. */
bool numeric::is_odd(void) const
{
- return (is_integer() &&
- oddp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN
}
/** Probabilistic primality test.
* @return true if object is exact integer and prime. */
bool numeric::is_prime(void) const
{
- return (is_integer() &&
- isprobprime(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN
}
/** True if object is an exact rational number, may even be complex
* (denominator may be unity). */
bool numeric::is_rational(void) const
{
- if (instanceof(*value, cl_RA_ring)) {
- return true;
- } else if (!is_real()) { // complex case, handle Q(i):
- if ( instanceof(realpart(*value), cl_RA_ring) &&
- instanceof(imagpart(*value), cl_RA_ring) )
- return true;
- }
- return false;
+ return ::instanceof(*value, ::cl_RA_ring); // -> CLN
}
/** True if object is a real integer, rational or float (but not complex). */
bool numeric::is_real(void) const
{
- return (bool)instanceof(*value, cl_R_ring); // -> CLN
+ return ::instanceof(*value, ::cl_R_ring); // -> CLN
}
-bool numeric::operator==(numeric const & other) const
+bool numeric::operator==(const numeric & other) const
{
return (*value == *other.value); // -> CLN
}
-bool numeric::operator!=(numeric const & other) const
+bool numeric::operator!=(const numeric & other) const
{
return (*value != *other.value); // -> CLN
}
+/** True if object is element of the domain of integers extended by I, i.e. is
+ * of the form a+b*I, where a and b are integers. */
+bool numeric::is_cinteger(void) const
+{
+ if (::instanceof(*value, ::cl_I_ring))
+ return true;
+ else if (!this->is_real()) { // complex case, handle n+m*I
+ if (::instanceof(::realpart(*value), ::cl_I_ring) &&
+ ::instanceof(::imagpart(*value), ::cl_I_ring))
+ return true;
+ }
+ return false;
+}
+
+/** True if object is an exact rational number, may even be complex
+ * (denominator may be unity). */
+bool numeric::is_crational(void) const
+{
+ if (::instanceof(*value, ::cl_RA_ring))
+ return true;
+ else if (!this->is_real()) { // complex case, handle Q(i):
+ if (::instanceof(::realpart(*value), ::cl_RA_ring) &&
+ ::instanceof(::imagpart(*value), ::cl_RA_ring))
+ return true;
+ }
+ return false;
+}
+
/** Numerical comparison: less.
*
* @exception invalid_argument (complex inequality) */
-bool numeric::operator<(numeric const & other) const
+bool numeric::operator<(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
- }
+ if (this->is_real() && other.is_real())
+ return (The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
throw (std::invalid_argument("numeric::operator<(): complex inequality"));
return false; // make compiler shut up
}
/** Numerical comparison: less or equal.
*
* @exception invalid_argument (complex inequality) */
-bool numeric::operator<=(numeric const & other) const
+bool numeric::operator<=(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
- }
+ if (this->is_real() && other.is_real())
+ return (The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
return false; // make compiler shut up
}
/** Numerical comparison: greater.
*
* @exception invalid_argument (complex inequality) */
-bool numeric::operator>(numeric const & other) const
+bool numeric::operator>(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
- }
+ if (this->is_real() && other.is_real())
+ return (The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
throw (std::invalid_argument("numeric::operator>(): complex inequality"));
return false; // make compiler shut up
}
/** Numerical comparison: greater or equal.
*
* @exception invalid_argument (complex inequality) */
-bool numeric::operator>=(numeric const & other) const
+bool numeric::operator>=(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
- }
+ if (this->is_real() && other.is_real())
+ return (The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
throw (std::invalid_argument("numeric::operator>=(): complex inequality"));
return false; // make compiler shut up
}
-/** Converts numeric types to machine's int. You should check with is_integer()
- * if the number is really an integer before calling this method. */
+/** Converts numeric types to machine's int. You should check with
+ * is_integer() if the number is really an integer before calling this method.
+ * You may also consider checking the range first. */
int numeric::to_int(void) const
{
- ASSERT(is_integer());
- return cl_I_to_int(The(cl_I)(*value));
+ GINAC_ASSERT(this->is_integer());
+ return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN
+}
+
+/** Converts numeric types to machine's long. You should check with
+ * is_integer() if the number is really an integer before calling this method.
+ * You may also consider checking the range first. */
+long numeric::to_long(void) const
+{
+ GINAC_ASSERT(this->is_integer());
+ return ::cl_I_to_long(The(cl_I)(*value)); // -> CLN
}
/** Converts numeric types to machine's double. You should check with is_real()
* if the number is really not complex before calling this method. */
double numeric::to_double(void) const
{
- ASSERT(is_real());
- return cl_double_approx(realpart(*value));
+ GINAC_ASSERT(this->is_real());
+ return ::cl_double_approx(::realpart(*value)); // -> CLN
}
/** Real part of a number. */
-numeric numeric::real(void) const
+const numeric numeric::real(void) const
{
- return numeric(realpart(*value)); // -> CLN
+ return numeric(::realpart(*value)); // -> CLN
}
/** Imaginary part of a number. */
-numeric numeric::imag(void) const
+const numeric numeric::imag(void) const
{
- return numeric(imagpart(*value)); // -> CLN
+ return numeric(::imagpart(*value)); // -> CLN
}
#ifndef SANE_LINKER
/** Numerator. Computes the numerator of rational numbers, rationalized
* numerator of complex if real and imaginary part are both rational numbers
- * (i.e numer(4/3+5/6*I) == 8+5*I), the number itself in all other cases. */
-numeric numeric::numer(void) const
+ * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
+ * cases. */
+const numeric numeric::numer(void) const
{
- if (is_integer()) {
+ if (this->is_integer()) {
return numeric(*this);
}
#ifdef SANE_LINKER
- else if (instanceof(*value, cl_RA_ring)) {
- return numeric(numerator(The(cl_RA)(*value)));
+ else if (::instanceof(*value, ::cl_RA_ring)) {
+ return numeric(::numerator(The(cl_RA)(*value)));
}
- else if (!is_real()) { // complex case, handle Q(i):
- cl_R r = realpart(*value);
- cl_R i = imagpart(*value);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
+ else if (!this->is_real()) { // complex case, handle Q(i):
+ cl_R r = ::realpart(*value);
+ cl_R i = ::imagpart(*value);
+ if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
return numeric(*this);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
- return numeric(complex(r*denominator(The(cl_RA)(i)), numerator(The(cl_RA)(i))));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
- return numeric(complex(numerator(The(cl_RA)(r)), i*denominator(The(cl_RA)(r))));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) {
- cl_I s = lcm(denominator(The(cl_RA)(r)), denominator(The(cl_RA)(i)));
- return numeric(complex(numerator(The(cl_RA)(r))*(exquo(s,denominator(The(cl_RA)(r)))),
- numerator(The(cl_RA)(i))*(exquo(s,denominator(The(cl_RA)(i))))));
+ if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
+ return numeric(::complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i))));
+ if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
+ return numeric(::complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r))));
+ if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring)) {
+ cl_I s = ::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)));
+ return numeric(::complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))),
+ ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i))))));
}
}
#else
- else if (instanceof(*value, cl_RA_ring)) {
+ else if (instanceof(*value, ::cl_RA_ring)) {
return numeric(TheRatio(*value)->numerator);
}
- else if (!is_real()) { // complex case, handle Q(i):
- cl_R r = realpart(*value);
- cl_R i = imagpart(*value);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
+ else if (!this->is_real()) { // complex case, handle Q(i):
+ cl_R r = ::realpart(*value);
+ cl_R i = ::imagpart(*value);
+ if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
return numeric(*this);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
- return numeric(complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
- return numeric(complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) {
- cl_I s = lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
- return numeric(complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
+ if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
+ return numeric(::complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
+ if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
+ return numeric(::complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
+ if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring)) {
+ cl_I s = ::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
+ return numeric(::complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
}
}
/** Denominator. Computes the denominator of rational numbers, common integer
* denominator of complex if real and imaginary part are both rational numbers
* (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
-numeric numeric::denom(void) const
+const numeric numeric::denom(void) const
{
- if (is_integer()) {
- return numONE();
+ if (this->is_integer()) {
+ return _num1();
}
#ifdef SANE_LINKER
- if (instanceof(*value, cl_RA_ring)) {
- return numeric(denominator(The(cl_RA)(*value)));
+ if (instanceof(*value, ::cl_RA_ring)) {
+ return numeric(::denominator(The(cl_RA)(*value)));
}
- if (!is_real()) { // complex case, handle Q(i):
- cl_R r = realpart(*value);
- cl_R i = imagpart(*value);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
- return numONE();
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
- return numeric(denominator(The(cl_RA)(i)));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
- return numeric(denominator(The(cl_RA)(r)));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring))
- return numeric(lcm(denominator(The(cl_RA)(r)), denominator(The(cl_RA)(i))));
+ if (!this->is_real()) { // complex case, handle Q(i):
+ cl_R r = ::realpart(*value);
+ cl_R i = ::imagpart(*value);
+ if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
+ return _num1();
+ if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
+ return numeric(::denominator(The(cl_RA)(i)));
+ if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
+ return numeric(::denominator(The(cl_RA)(r)));
+ if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring))
+ return numeric(::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
}
#else
- if (instanceof(*value, cl_RA_ring)) {
+ if (instanceof(*value, ::cl_RA_ring)) {
return numeric(TheRatio(*value)->denominator);
}
- if (!is_real()) { // complex case, handle Q(i):
- cl_R r = realpart(*value);
- cl_R i = imagpart(*value);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
- return numONE();
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
+ if (!this->is_real()) { // complex case, handle Q(i):
+ cl_R r = ::realpart(*value);
+ cl_R i = ::imagpart(*value);
+ if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
+ return _num1();
+ if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
return numeric(TheRatio(i)->denominator);
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
+ if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
return numeric(TheRatio(r)->denominator);
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring))
- return numeric(lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
+ if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring))
+ return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
}
#endif // def SANE_LINKER
// at least one float encountered
- return numONE();
+ return _num1();
}
/** Size in binary notation. For integers, this is the smallest n >= 0 such
* in two's complement if it is an integer, 0 otherwise. */
int numeric::int_length(void) const
{
- if (is_integer()) {
- return integer_length(The(cl_I)(*value)); // -> CLN
- } else {
+ if (this->is_integer())
+ return ::integer_length(The(cl_I)(*value)); // -> CLN
+ else
return 0;
- }
}
//////////
const numeric some_numeric;
-type_info const & typeid_numeric=typeid(some_numeric);
+const type_info & typeid_numeric=typeid(some_numeric);
/** Imaginary unit. This is not a constant but a numeric since we are
* natively handing complex numbers anyways. */
-const numeric I = (complex(cl_I(0),cl_I(1)));
-
-//////////
-// global functions
-//////////
-
-numeric const & numZERO(void)
-{
- const static ex eZERO = ex((new numeric(0))->setflag(status_flags::dynallocated));
- const static numeric * nZERO = static_cast<const numeric *>(eZERO.bp);
- return *nZERO;
-}
+const numeric I = numeric(::complex(cl_I(0),cl_I(1)));
-numeric const & numONE(void)
-{
- const static ex eONE = ex((new numeric(1))->setflag(status_flags::dynallocated));
- const static numeric * nONE = static_cast<const numeric *>(eONE.bp);
- return *nONE;
-}
-
-numeric const & numTWO(void)
-{
- const static ex eTWO = ex((new numeric(2))->setflag(status_flags::dynallocated));
- const static numeric * nTWO = static_cast<const numeric *>(eTWO.bp);
- return *nTWO;
-}
-
-numeric const & numTHREE(void)
-{
- const static ex eTHREE = ex((new numeric(3))->setflag(status_flags::dynallocated));
- const static numeric * nTHREE = static_cast<const numeric *>(eTHREE.bp);
- return *nTHREE;
-}
-
-numeric const & numMINUSONE(void)
-{
- const static ex eMINUSONE = ex((new numeric(-1))->setflag(status_flags::dynallocated));
- const static numeric * nMINUSONE = static_cast<const numeric *>(eMINUSONE.bp);
- return *nMINUSONE;
-}
-
-numeric const & numHALF(void)
-{
- const static ex eHALF = ex((new numeric(1, 2))->setflag(status_flags::dynallocated));
- const static numeric * nHALF = static_cast<const numeric *>(eHALF.bp);
- return *nHALF;
-}
/** Exponential function.
*
* @return arbitrary precision numerical exp(x). */
-numeric exp(numeric const & x)
+const numeric exp(const numeric & x)
{
- return exp(*x.value); // -> CLN
+ return ::exp(*x.value); // -> CLN
}
+
/** Natural logarithm.
*
* @param z complex number
* @return arbitrary precision numerical log(x).
* @exception overflow_error (logarithmic singularity) */
-numeric log(numeric const & z)
+const numeric log(const numeric & z)
{
if (z.is_zero())
throw (std::overflow_error("log(): logarithmic singularity"));
- return log(*z.value); // -> CLN
+ return ::log(*z.value); // -> CLN
}
+
/** Numeric sine (trigonometric function).
*
* @return arbitrary precision numerical sin(x). */
-numeric sin(numeric const & x)
+const numeric sin(const numeric & x)
{
- return sin(*x.value); // -> CLN
+ return ::sin(*x.value); // -> CLN
}
+
/** Numeric cosine (trigonometric function).
*
* @return arbitrary precision numerical cos(x). */
-numeric cos(numeric const & x)
+const numeric cos(const numeric & x)
{
- return cos(*x.value); // -> CLN
+ return ::cos(*x.value); // -> CLN
}
-
+
+
/** Numeric tangent (trigonometric function).
*
* @return arbitrary precision numerical tan(x). */
-numeric tan(numeric const & x)
+const numeric tan(const numeric & x)
{
- return tan(*x.value); // -> CLN
+ return ::tan(*x.value); // -> CLN
}
+
/** Numeric inverse sine (trigonometric function).
*
* @return arbitrary precision numerical asin(x). */
-numeric asin(numeric const & x)
+const numeric asin(const numeric & x)
{
- return asin(*x.value); // -> CLN
+ return ::asin(*x.value); // -> CLN
}
-
+
+
/** Numeric inverse cosine (trigonometric function).
*
* @return arbitrary precision numerical acos(x). */
-numeric acos(numeric const & x)
+const numeric acos(const numeric & x)
{
- return acos(*x.value); // -> CLN
+ return ::acos(*x.value); // -> CLN
}
-/** Arcustangents.
+
+/** Arcustangent.
*
* @param z complex number
* @return atan(z)
* @exception overflow_error (logarithmic singularity) */
-numeric atan(numeric const & x)
+const numeric atan(const numeric & x)
{
if (!x.is_real() &&
x.real().is_zero() &&
- !abs(x.imag()).is_equal(numONE()))
+ !abs(x.imag()).is_equal(_num1()))
throw (std::overflow_error("atan(): logarithmic singularity"));
- return atan(*x.value); // -> CLN
+ return ::atan(*x.value); // -> CLN
}
-/** Arcustangents.
+
+/** Arcustangent.
*
* @param x real number
* @param y real number
* @return atan(y/x) */
-numeric atan(numeric const & y, numeric const & x)
+const numeric atan(const numeric & y, const numeric & x)
{
if (x.is_real() && y.is_real())
- return atan(realpart(*x.value), realpart(*y.value)); // -> CLN
+ return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
else
throw (std::invalid_argument("numeric::atan(): complex argument"));
}
+
/** Numeric hyperbolic sine (trigonometric function).
*
* @return arbitrary precision numerical sinh(x). */
-numeric sinh(numeric const & x)
+const numeric sinh(const numeric & x)
{
- return sinh(*x.value); // -> CLN
+ return ::sinh(*x.value); // -> CLN
}
+
/** Numeric hyperbolic cosine (trigonometric function).
*
* @return arbitrary precision numerical cosh(x). */
-numeric cosh(numeric const & x)
+const numeric cosh(const numeric & x)
{
- return cosh(*x.value); // -> CLN
+ return ::cosh(*x.value); // -> CLN
}
-
+
+
/** Numeric hyperbolic tangent (trigonometric function).
*
* @return arbitrary precision numerical tanh(x). */
-numeric tanh(numeric const & x)
+const numeric tanh(const numeric & x)
{
- return tanh(*x.value); // -> CLN
+ return ::tanh(*x.value); // -> CLN
}
+
/** Numeric inverse hyperbolic sine (trigonometric function).
*
* @return arbitrary precision numerical asinh(x). */
-numeric asinh(numeric const & x)
+const numeric asinh(const numeric & x)
{
- return asinh(*x.value); // -> CLN
+ return ::asinh(*x.value); // -> CLN
}
+
/** Numeric inverse hyperbolic cosine (trigonometric function).
*
* @return arbitrary precision numerical acosh(x). */
-numeric acosh(numeric const & x)
+const numeric acosh(const numeric & x)
{
- return acosh(*x.value); // -> CLN
+ return ::acosh(*x.value); // -> CLN
}
+
/** Numeric inverse hyperbolic tangent (trigonometric function).
*
* @return arbitrary precision numerical atanh(x). */
-numeric atanh(numeric const & x)
+const numeric atanh(const numeric & x)
{
- return atanh(*x.value); // -> CLN
+ return ::atanh(*x.value); // -> CLN
}
-/** The gamma function.
- * stub stub stub stub stub stub! */
-numeric gamma(numeric const & x)
+
+/** Numeric evaluation of Riemann's Zeta function. Currently works only for
+ * integer arguments. */
+const numeric zeta(const numeric & x)
{
- clog << "gamma(): Nobody expects the Spanish inquisition" << endl;
+ // A dirty hack to allow for things like zeta(3.0), since CLN currently
+ // only knows about integer arguments and zeta(3).evalf() automatically
+ // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
+ // being an exact zero for CLN, which can be tested and then we can just
+ // pass the number casted to an int:
+ if (x.is_real()) {
+ int aux = (int)(::cl_double_approx(::realpart(*x.value)));
+ if (zerop(*x.value-aux))
+ return ::cl_zeta(aux); // -> CLN
+ }
+ clog << "zeta(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << endl;
return numeric(0);
}
+
+/** The Gamma function.
+ * This is only a stub! */
+const numeric lgamma(const numeric & x)
+{
+ clog << "lgamma(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << endl;
+ return numeric(0);
+}
+const numeric tgamma(const numeric & x)
+{
+ clog << "tgamma(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << endl;
+ return numeric(0);
+}
+
+
+/** The psi function (aka polygamma function).
+ * This is only a stub! */
+const numeric psi(const numeric & x)
+{
+ clog << "psi(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << endl;
+ return numeric(0);
+}
+
+
+/** The psi functions (aka polygamma functions).
+ * This is only a stub! */
+const numeric psi(const numeric & n, const numeric & x)
+{
+ clog << "psi(" << n << "," << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << endl;
+ return numeric(0);
+}
+
+
/** Factorial combinatorial function.
*
+ * @param n integer argument >= 0
* @exception range_error (argument must be integer >= 0) */
-numeric factorial(numeric const & nn)
+const numeric factorial(const numeric & n)
{
- if ( !nn.is_nonneg_integer() ) {
+ if (!n.is_nonneg_integer())
throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
- }
-
- return numeric(factorial(nn.to_int())); // -> CLN
+ return numeric(::factorial(n.to_int())); // -> CLN
}
+
/** The double factorial combinatorial function. (Scarcely used, but still
- * useful in cases, like for exact results of Gamma(n+1/2) for instance.)
+ * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
*
* @param n integer argument >= -1
- * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == 1 == (-1)!!
+ * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
* @exception range_error (argument must be integer >= -1) */
-numeric doublefactorial(numeric const & nn)
-{
- // We store the results separately for even and odd arguments. This has
- // the advantage that we don't have to compute any even result at all if
- // the function is always called with odd arguments and vice versa. There
- // is no tradeoff involved in this, it is guaranteed to save time as well
- // as memory. (If this is not enough justification consider the Gamma
- // function of half integer arguments: it only needs odd doublefactorials.)
- static vector<numeric> evenresults;
- static int highest_evenresult = -1;
- static vector<numeric> oddresults;
- static int highest_oddresult = -1;
-
- if ( nn == numeric(-1) ) {
- return numONE();
+const numeric doublefactorial(const numeric & n)
+{
+ if (n == numeric(-1)) {
+ return _num1();
}
- if ( !nn.is_nonneg_integer() ) {
+ if (!n.is_nonneg_integer()) {
throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
}
- if ( nn.is_even() ) {
- int n = nn.div(numTWO()).to_int();
- if ( n <= highest_evenresult ) {
- return evenresults[n];
- }
- if ( evenresults.capacity() < (unsigned)(n+1) ) {
- evenresults.reserve(n+1);
- }
- if ( highest_evenresult < 0 ) {
- evenresults.push_back(numONE());
- highest_evenresult=0;
- }
- for (int i=highest_evenresult+1; i<=n; i++) {
- evenresults.push_back(numeric(evenresults[i-1].mul(numeric(i*2))));
- }
- highest_evenresult=n;
- return evenresults[n];
- } else {
- int n = nn.sub(numONE()).div(numTWO()).to_int();
- if ( n <= highest_oddresult ) {
- return oddresults[n];
- }
- if ( oddresults.capacity() < (unsigned)n ) {
- oddresults.reserve(n+1);
- }
- if ( highest_oddresult < 0 ) {
- oddresults.push_back(numONE());
- highest_oddresult=0;
- }
- for (int i=highest_oddresult+1; i<=n; i++) {
- oddresults.push_back(numeric(oddresults[i-1].mul(numeric(i*2+1))));
+ return numeric(::doublefactorial(n.to_int())); // -> CLN
+}
+
+
+/** The Binomial coefficients. It computes the binomial coefficients. For
+ * integer n and k and positive n this is the number of ways of choosing k
+ * objects from n distinct objects. If n is negative, the formula
+ * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
+const numeric binomial(const numeric & n, const numeric & k)
+{
+ if (n.is_integer() && k.is_integer()) {
+ if (n.is_nonneg_integer()) {
+ if (k.compare(n)!=1 && k.compare(_num0())!=-1)
+ return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
+ else
+ return _num0();
+ } else {
+ return _num_1().power(k)*binomial(k-n-_num1(),k);
}
- highest_oddresult=n;
- return oddresults[n];
}
+
+ // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit
+ throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
}
-/** The Binomial function. It computes the binomial coefficients. If the
- * arguments are both nonnegative integers and 0 <= k <= n, then
- * binomial(n, k) = n!/k!/(n-k)! which is the number of ways of choosing k
- * objects from n distinct objects. If k > n, then binomial(n,k) returns 0. */
-numeric binomial(numeric const & n, numeric const & k)
-{
- if (n.is_nonneg_integer() && k.is_nonneg_integer()) {
- return numeric(binomial(n.to_int(),k.to_int())); // -> CLN
- } else {
- // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1)
- return numeric(0);
+
+/** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
+ * in the expansion of the function x/(e^x-1).
+ *
+ * @return the nth Bernoulli number (a rational number).
+ * @exception range_error (argument must be integer >= 0) */
+const numeric bernoulli(const numeric & nn)
+{
+ if (!nn.is_integer() || nn.is_negative())
+ throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
+ if (nn.is_zero())
+ return _num1();
+ if (!nn.compare(_num1()))
+ return numeric(-1,2);
+ if (nn.is_odd())
+ return _num0();
+ // Until somebody has the blues and comes up with a much better idea and
+ // codes it (preferably in CLN) we make this a remembering function which
+ // computes its results using the defining formula
+ // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
+ // whith B(0) == 1.
+ // Be warned, though: the Bernoulli numbers are computationally very
+ // expensive anyhow and you shouldn't expect miracles to happen.
+ static vector<numeric> results;
+ static int highest_result = -1;
+ int n = nn.sub(_num2()).div(_num2()).to_int();
+ if (n <= highest_result)
+ return results[n];
+ if (results.capacity() < (unsigned)(n+1))
+ results.reserve(n+1);
+
+ numeric tmp; // used to store the sum
+ for (int i=highest_result+1; i<=n; ++i) {
+ // the first two elements:
+ tmp = numeric(-2*i-1,2);
+ // accumulate the remaining elements:
+ for (int j=0; j<i; ++j)
+ tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
+ // divide by -(nn+1) and store result:
+ results.push_back(-tmp/numeric(2*i+3));
}
- // return factorial(n).div(factorial(k).mul(factorial(n.sub(k))));
+ highest_result=n;
+ return results[n];
}
+
+/** Fibonacci number. The nth Fibonacci number F(n) is defined by the
+ * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
+ *
+ * @param n an integer
+ * @return the nth Fibonacci number F(n) (an integer number)
+ * @exception range_error (argument must be an integer) */
+const numeric fibonacci(const numeric & n)
+{
+ if (!n.is_integer())
+ throw (std::range_error("numeric::fibonacci(): argument must be integer"));
+ // The following addition formula holds:
+ // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
+ // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
+ // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
+ // agree.)
+ // Replace m by m+1:
+ // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
+ // Now put in m = n, to get
+ // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
+ // F(2n+1) = F(n)^2 + F(n+1)^2
+ // hence
+ // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
+ if (n.is_zero())
+ return _num0();
+ if (n.is_negative())
+ if (n.is_even())
+ return -fibonacci(-n);
+ else
+ return fibonacci(-n);
+
+ cl_I u(0);
+ cl_I v(1);
+ cl_I m = The(cl_I)(*n.value) >> 1L; // floor(n/2);
+ for (uintL bit=::integer_length(m); bit>0; --bit) {
+ // Since a squaring is cheaper than a multiplication, better use
+ // three squarings instead of one multiplication and two squarings.
+ cl_I u2 = ::square(u);
+ cl_I v2 = ::square(v);
+ if (::logbitp(bit-1, m)) {
+ v = ::square(u + v) - u2;
+ u = u2 + v2;
+ } else {
+ u = v2 - ::square(v - u);
+ v = u2 + v2;
+ }
+ }
+ if (n.is_even())
+ // Here we don't use the squaring formula because one multiplication
+ // is cheaper than two squarings.
+ return u * ((v << 1) - u);
+ else
+ return ::square(u) + ::square(v);
+}
+
+
/** Absolute value. */
-numeric abs(numeric const & x)
+numeric abs(const numeric & x)
{
- return abs(*x.value); // -> CLN
+ return ::abs(*x.value); // -> CLN
}
+
/** Modulus (in positive representation).
* In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
* sign of a or is zero. This is different from Maple's modp, where the sign
*
* @return a mod b in the range [0,abs(b)-1] with sign of b if both are
* integer, 0 otherwise. */
-numeric mod(numeric const & a, numeric const & b)
+numeric mod(const numeric & a, const numeric & b)
{
- if (a.is_integer() && b.is_integer()) {
- return mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- }
- else {
- return numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer())
+ return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
+ else
+ return _num0(); // Throw?
}
+
/** Modulus (in symmetric representation).
* Equivalent to Maple's mods.
*
* @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
-numeric smod(numeric const & a, numeric const & b)
+numeric smod(const numeric & a, const numeric & b)
{
if (a.is_integer() && b.is_integer()) {
cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1;
- return mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
- } else {
- return numZERO(); // Throw?
- }
+ return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
+ } else
+ return _num0(); // Throw?
}
+
/** Numeric integer remainder.
* Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
* In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
* sign of a or is zero.
*
* @return remainder of a/b if both are integer, 0 otherwise. */
-numeric irem(numeric const & a, numeric const & b)
+numeric irem(const numeric & a, const numeric & b)
{
- if (a.is_integer() && b.is_integer()) {
- return rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- }
- else {
- return numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer())
+ return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
+ else
+ return _num0(); // Throw?
}
+
/** Numeric integer remainder.
* Equivalent to Maple's irem(a,b,'q') it obeyes the relation
* irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
*
* @return remainder of a/b and quotient stored in q if both are integer,
* 0 otherwise. */
-numeric irem(numeric const & a, numeric const & b, numeric & q)
+numeric irem(const numeric & a, const numeric & b, numeric & q)
{
if (a.is_integer() && b.is_integer()) { // -> CLN
cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
return rem_quo.remainder;
}
else {
- q = numZERO();
- return numZERO(); // Throw?
+ q = _num0();
+ return _num0(); // Throw?
}
}
+
/** Numeric integer quotient.
* Equivalent to Maple's iquo as far as sign conventions are concerned.
*
* @return truncated quotient of a/b if both are integer, 0 otherwise. */
-numeric iquo(numeric const & a, numeric const & b)
+numeric iquo(const numeric & a, const numeric & b)
{
- if (a.is_integer() && b.is_integer()) {
+ if (a.is_integer() && b.is_integer())
return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- } else {
- return numZERO(); // Throw?
- }
+ else
+ return _num0(); // Throw?
}
+
/** Numeric integer quotient.
* Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
* r == a - iquo(a,b,r)*b.
*
* @return truncated quotient of a/b and remainder stored in r if both are
* integer, 0 otherwise. */
-numeric iquo(numeric const & a, numeric const & b, numeric & r)
+numeric iquo(const numeric & a, const numeric & b, numeric & r)
{
if (a.is_integer() && b.is_integer()) { // -> CLN
cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
r = rem_quo.remainder;
return rem_quo.quotient;
} else {
- r = numZERO();
- return numZERO(); // Throw?
+ r = _num0();
+ return _num0(); // Throw?
}
}
+
/** Numeric square root.
* If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
* should return integer 2.
* @return square root of z. Branch cut along negative real axis, the negative
* real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
* where imag(z)>0. */
-numeric sqrt(numeric const & z)
+numeric sqrt(const numeric & z)
{
- return sqrt(*z.value); // -> CLN
+ return ::sqrt(*z.value); // -> CLN
}
+
/** Integer numeric square root. */
-numeric isqrt(numeric const & x)
+numeric isqrt(const numeric & x)
{
- if (x.is_integer()) {
- cl_I root;
- isqrt(The(cl_I)(*x.value), &root); // -> CLN
- return root;
- } else
- return numZERO(); // Throw?
+ if (x.is_integer()) {
+ cl_I root;
+ ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
+ return root;
+ } else
+ return _num0(); // Throw?
}
+
/** Greatest Common Divisor.
*
* @return The GCD of two numbers if both are integer, a numerical 1
* if they are not. */
-numeric gcd(numeric const & a, numeric const & b)
+numeric gcd(const numeric & a, const numeric & b)
{
if (a.is_integer() && b.is_integer())
- return gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
+ return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
else
- return numONE();
+ return _num1();
}
+
/** Least Common Multiple.
*
* @return The LCM of two numbers if both are integer, the product of those
* two numbers if they are not. */
-numeric lcm(numeric const & a, numeric const & b)
+numeric lcm(const numeric & a, const numeric & b)
{
if (a.is_integer() && b.is_integer())
- return lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
+ return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
else
return *a.value * *b.value;
}
+
+/** Floating point evaluation of Archimedes' constant Pi. */
ex PiEvalf(void)
{
- return numeric(cl_pi(cl_default_float_format)); // -> CLN
+ return numeric(::cl_pi(cl_default_float_format)); // -> CLN
}
-ex EulerGammaEvalf(void)
+
+/** Floating point evaluation of Euler's constant gamma. */
+ex EulerEvalf(void)
{
- return numeric(cl_eulerconst(cl_default_float_format)); // -> CLN
+ return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
}
+
+/** Floating point evaluation of Catalan's constant. */
ex CatalanEvalf(void)
{
- return numeric(cl_catalanconst(cl_default_float_format)); // -> CLN
+ return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
}
+
// It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
// be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
// of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
{
assert(!too_late);
too_late = true;
- cl_default_float_format = cl_float_format(17);
+ cl_default_float_format = ::cl_float_format(17);
}
+
_numeric_digits& _numeric_digits::operator=(long prec)
{
digits=prec;
- cl_default_float_format = cl_float_format(prec);
+ cl_default_float_format = ::cl_float_format(prec);
return *this;
}
+
_numeric_digits::operator long()
{
return (long)digits;
}
+
void _numeric_digits::print(ostream & os) const
{
debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
os << digits;
}
-ostream& operator<<(ostream& os, _numeric_digits const & e)
+
+ostream& operator<<(ostream& os, const _numeric_digits & e)
{
e.print(os);
return os;
bool _numeric_digits::too_late = false;
+
/** Accuracy in decimal digits. Only object of this type! Can be set using
* assignment from C++ unsigned ints and evaluated like any built-in type. */
_numeric_digits Digits;
+
+#ifndef NO_NAMESPACE_GINAC
+} // namespace GiNaC
+#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff